I seem to be unable to make any changes to the settings. For example, when I activate timestamps for chat, it doesn't work. When I open "Settings" again, it is unchecked. Same goes for every other setting. And game groups.

This is something like a corrupt localconfig.vdf file, it should be in Steam/userdata/*/config/. If you make a backup of that file and then delete it and restart steam, does this issue (and #2282) fix itself?